A229870 -id:A229870 - cp's OEIS frontend

A229865 Number of n X n 0..1 arrays with corresponding row and column sums equal.

1, 2, 8, 80, 2432, 247552, 88060928, 112371410944, 523858015518720, 9041009511609073664, 583447777113052431515648, 141885584718620229407228821504, 130832005909904417592540055577034752, 459749137931232137234615429529864283095040, 6182706200522446492946534924719926752508110700544

Offset: 0

R. H. Hardin, Oct 01 2013

nonn

Also known as labeled Eulerian digraphs allowing loops. - Brendan McKay, May 12 2019

			Some solutions for n=4:
  0 0 0 1     0 0 1 0     0 0 0 1     0 0 1 0     0 0 1 1
  0 1 0 0     1 0 0 0     1 0 1 0     0 0 1 1     1 0 0 1
  0 0 0 1     0 1 0 0     0 1 0 1     0 1 1 1     1 1 1 0
  1 0 1 0     0 0 0 1     0 1 1 0     1 1 0 0     0 1 1 1
From _Gus Wiseman_, Jun 22 2019: (Start)
The a(3) = 8 Eulerian digraph edge-sets:
  {}
  {11}
  {22}
  {11,22}
  {12,21}
  {11,12,21}
  {12,21,22}
  {11,12,21,22}
(End)

Mohammad Behzad Kang and Andrew Salch, The mod p cohomology of the Morava stabilizer group at large primes, arXiv:2410.24171 [math.AT], 2024. See p. 46.

Column 1 of A229870.
The unlabeled version is A308111.
Cf. A000595, A002416, A002720, A007080.
Cf. A326209, A326237, A326251, A326252, A326253.

Table[Length[Select[Subsets[Tuples[Range[n],2]],Sort[First/@#]==Sort[Last/@#]&]],{n,4}] (* Gus Wiseman, Jun 22 2019 *)

a(n) = 2^n * A007080(n). - Andrew Howroyd, Sep 11 2019

a(0)=1 prepended by Alois P. Heinz, May 14 2019

Terms a(11) and beyond from Andrew Howroyd, Sep 11 2019

A229417 T(n,k) = number of n X n 0..k zero-diagonal arrays with corresponding row and column sums equal.

Original entry on oeis.org

1, 1, 2, 1, 3, 10, 1, 4, 45, 152, 1, 5, 136, 4743, 7736, 1, 6, 325, 59008, 3801411, 1375952, 1, 7, 666, 426425, 345706336, 23938685973, 877901648, 1, 8, 1225, 2164680, 11782824375, 28256240134144, 1215663478473627, 2046320373120, 1, 9, 2080

Offset: 1

R. H. Hardin, Sep 22 2013

Table starts
.........1................1....................1................1............1
.........2................3....................4................5............6
........10...............45..................136..............325..........666
.......152.............4743................59008...........426425......2164680
......7736..........3801411............345706336......11782824375.213067487016
...1375952......23938685973.......28256240134144.7093199984236625
.877901648.1215663478473627.33097994593655140864

			Some solutions for n=4 k=4
..0..0..2..0....0..1..0..4....0..0..1..3....0..1..1..4....0..1..1..0
..1..0..2..1....2..0..4..0....1..0..2..3....4..0..2..3....0..0..1..2
..1..2..0..4....2..4..0..2....2..3..0..1....1..4..0..1....0..0..0..4
..0..2..3..0....1..1..4..0....1..3..3..0....1..4..3..0....2..2..2..0

R. H. Hardin, Table of n, a(n) for n = 1..43

Columns 1..3 are A007080, A229415, A229416.
Rows 3..6 are A037270(n+1), A229418, A229419, A229420.
Cf. A229870.

Empirical for row n:
n=1: a(n) = 1
n=2: a(n) = n + 1
n=3: a(n) = (1/2)*n^4 + 2*n^3 + (7/2)*n^2 + 3*n + 1
n=4: [polynomial of degree 9]
Row n is an Ehrhart polynomial of degree (n-1)^2 for the polytope of x(i,j), i,j = 1..n for j <> i, with 0 <= x(i,j) <= 1 and Sum_i x(i,j) = Sum_i x(j,i). - Robert Israel, Mar 30 2023
T(n,k) = A229870(n,k) / (k + 1)^n. - Andrew Howroyd, Mar 30 2023

A229871 Number of 4X4 0..n arrays with corresponding row and column sums equal.

Original entry on oeis.org

2432, 384183, 15106048, 266515625, 2805425280, 20610104767, 116211580928, 535021318017, 2098414000000, 7227967108439, 22362655260672, 63219695614153, 165498794685056, 405459218109375, 937601561264128, 2060839403682497

Offset: 1

R. H. Hardin, Oct 01 2013

nonn

Row 4 of A229870

			Some solutions for n=4
..0..0..0..0....0..0..0..1....0..0..0..1....0..0..1..1....0..0..1..1
..0..1..2..3....0..0..4..2....0..1..1..3....0..1..2..1....0..1..0..2
..0..4..0..0....1..4..0..1....1..4..2..0....1..3..4..3....1..2..0..0
..0..1..2..3....0..2..2..3....0..0..4..3....1..0..4..1....1..0..2..0

R. H. Hardin, Table of n, a(n) for n = 1..73

Empirical: a(n) = (29/140)*n^13 + (377/140)*n^12 + (3449/210)*n^11 + (13057/210)*n^10 + (34249/210)*n^9 + (21941/70)*n^8 + (6826/15)*n^7 + (53206/105)*n^6 + (60773/140)*n^5 + (119003/420)*n^4 + (28807/210)*n^3 + (9791/210)*n^2 + (348/35)*n + 1

A229872 Number of 5X5 0..n arrays with corresponding row and column sums equal.

Original entry on oeis.org

247552, 923742873, 354003288064, 36821326171875, 1656812779036416, 41629349513286507, 681576779395104768, 8038582007461068429, 73160330136500000000, 539739832319262606581, 3347559386080200916992

Offset: 1

R. H. Hardin, Oct 01 2013

nonn

Row 5 of A229870

			Some solutions for n=4
..0..0..0..0..1....0..0..0..0..0....0..0..0..0..1....0..0..0..0..0
..0..0..0..0..0....0..0..0..0..0....0..0..0..0..0....0..0..0..0..0
..0..0..3..0..4....0..0..3..2..3....0..0..1..2..4....0..0..0..1..3
..0..0..1..2..3....0..0..4..3..2....0..0..4..0..1....0..0..2..2..3
..1..0..3..4..2....0..0..1..4..1....1..0..2..3..1....0..0..2..4..3

R. H. Hardin, Table of n, a(n) for n = 1..15

A229866 Number of n X n 0..2 arrays with corresponding row and column sums equal.

Original entry on oeis.org

3, 27, 1215, 384183, 923742873, 17451302074317, 2658656027421822249

Offset: 1

R. H. Hardin, Oct 01 2013

nonn

Column 2 of A229870

			Some solutions for n=4
..0..0..1..2....0..1..2..1....0..0..2..2....0..1..0..0....0..0..1..1
..1..1..1..0....2..2..0..2....2..1..0..0....0..2..0..1....1..2..0..1
..0..2..1..0....2..2..1..0....2..2..0..0....1..0..2..1....1..0..0..2
..2..0..0..2....0..1..2..2....0..0..2..1....0..0..2..1....0..2..2..1

A229867 Number of n X n 0..3 arrays with corresponding row and column sums equal.

Original entry on oeis.org

4, 64, 8704, 15106048, 354003288064, 115737559589453824, 542277543422445827915776

Offset: 1

R. H. Hardin, Oct 01 2013

nonn

Column 3 of A229870

			Some solutions for n=4
..0..0..2..2....0..2..0..2....0..0..2..2....0..2..0..0....0..2..2..0
..2..1..0..2....0..0..2..0....0..2..3..2....2..3..3..0....2..1..1..0
..1..2..1..2....3..0..2..1....3..2..1..1....0..0..1..3....2..0..3..2
..1..2..3..2....1..0..2..0....1..3..1..0....0..3..0..0....0..1..1..1

A229868 Number of n X n 0..4 arrays with corresponding row and column sums equal.

Original entry on oeis.org

5, 125, 40625, 266515625, 36821326171875, 110831249753697265625

Offset: 1

R. H. Hardin, Oct 01 2013

nonn

Column 4 of A229870

			Some solutions for n=4
..0..0..0..1....0..0..1..1....0..0..1..0....0..0..1..0....0..0..1..0
..0..0..2..1....0..0..2..1....0..0..1..2....0..0..2..0....0..0..4..1
..1..2..4..0....2..2..3..2....1..0..2..4....0..2..2..4....1..4..0..0
..0..1..1..0....0..1..3..0....0..3..3..4....1..0..3..2....0..1..0..0

A229869 Number of n X n 0..5 arrays with corresponding row and column sums equal.

Original entry on oeis.org

6, 216, 143856, 2805425280, 1656812779036416, 30688414550520167402496

Offset: 1

R. H. Hardin, Oct 01 2013

nonn

Column 5 of A229870

			Some solutions for n=4
..0..0..3..3....0..3..0..3....0..3..3..0....0..3..3..3....0..0..3..3
..3..0..3..0....3..0..0..0....3..0..0..3....3..0..3..3....3..0..0..3
..1..4..2..5....1..0..2..4....2..1..0..3....2..2..3..2....2..4..3..1
..2..2..4..2....2..0..5..5....1..2..3..0....4..4..0..1....1..2..4..0

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A229865 Number of n X n 0..1 arrays with corresponding row and column sums equal.

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A229417 T(n,k) = number of n X n 0..k zero-diagonal arrays with corresponding row and column sums equal.

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A229871 Number of 4X4 0..n arrays with corresponding row and column sums equal.

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A229872 Number of 5X5 0..n arrays with corresponding row and column sums equal.

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A229866 Number of n X n 0..2 arrays with corresponding row and column sums equal.

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A229867 Number of n X n 0..3 arrays with corresponding row and column sums equal.

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A229868 Number of n X n 0..4 arrays with corresponding row and column sums equal.

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A229869 Number of n X n 0..5 arrays with corresponding row and column sums equal.

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